2 The Technique Lia Ca 
The component « is observed in a cuboidal lattice of points, i.e. we consider spatial digital the cell ed 
images of the microstructure. The discrete version of a (random) set forms a (random) binary open foam 
digital image. Depending on whether a lattice point is in a or in its complementary set, this ERAS 
point is assigned the Boolean values 1 or 0, respectively. The selection can be performed e.g. that form 
by thresholding brightness values to separate the a-phase from the background. or the exp 
Procedures for estimating the quermassdensities are based on Crofton’s intersection formu- geh ME TO 
lae, [1, p. 235], as well as a modification of Hadwiger’s recursive definition of the Euler AER 
number, [3], [4], [5, §2.4]. For the purpose of application in image analysis, the integrals In the cas 
that occur in the Crofton’s intersection formulae and Hadwiger’s recursive definition are dis- estimated 
cretized in such a way that ‘measurement’ of the geometric characteristics can be performed eroded wi 
by simple ‘counting’ of elements in a digital image where the elements are voxels or neigh- of the col 
borhood configurations of voxels. The observation of the structure in a point lattice implies Eo 
a corresponding discretization of the integral-geometric formulae. fon 
The statistical estimation of the quermassdensities suggested in the following includes linear 
filtering of the binary image as a basic tool. The algorithm presented in this paper consists 
of three steps: 
1. Filtering of the binary image (the ‘labeling of neighborhood configurations’ in the 
binary image), 
2. generating the vector of the numbers of neighborhood configurations (the ‘integration 
step’), and 
3. estimating the geometric characteristics from the absolute frequencies of configurations 
(the ‘analysis step’). 
By means of filtering, each neighborhood configuration in a binary image is assigned an 
integer. Thus, the result of the filtering is an image of integer valued voxels. The generation 
of the absolute frequencies of configurations can be understood as a discretized version of 
the computation of the translative integrals occurring in Crofton’s intersection formulae, and 
the vector of absolute frequencies carries the ‘complete information’ of the image about the 
quermassdensities; it can be used as the data base of statistical estimation. Since the neigh- i 
borhood configurations are represented by ‘grey-tones’, the vector of absolute frequencies of 
the neighborhood configuration in the binary image is nothing other than the vector of the Do 
absolute frequencies of grey-tones in the filtered image. In [5, §4| this technique has been 
described in all details where one can also find efficient source codes of C functions which 
compute the basic characteristics from the 3d voxel matrix. 
3 Application 
Of course, in practical application of materials characterization the last two basic charac- 
teristics, i.e. the densities of the two curvature integrals My and Ky, are not yet used very 
often. The reason for this is that their meaning is not obvious and a clear geometric inter- 
pretation of My and Ky can be given only for special types of microstructures. Examples 
are given bv the following two applications. S 
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